Question

Which equations represent hyperbolas?

Answer 1

Answer:

The correct options are 2 and 5.

Step-by-step explanation:

The general form of conics is

$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

This equation represents the hyperbola if

$B^2-4AC>0$

compare the given equations with the general equation.

In option 1,

$A=2,B=0,C=2,D=16,E=14,F=-9$

$B^2-4AC=(0)^2-4(2)(2)=-16$

This equation does not represents a hyperbola.

In option 2,

$A=2,B=0,C=-5,D=4,E=-10,F=57$

$B^2-4AC=(0)^2-4(2)(-5)=40>0$

This equation represents a hyperbola.

In option 3,

$A=-1,B=0,C=-7,D=5,E=2,F=-81$

$B^2-4AC=(0)^2-4(-1)(-7)=-28$

This equation does not represents a hyperbola.

In option 4,

$A=0,B=0,C=-2,D=1,E=4,F=15$

$B^2-4AC=(0)^2-4(0)(-2)=0$

This equation does not represents a hyperbola.

In option 5,

$A=-1,B=0,C=3,D=12,E=7,F=11$

$B^2-4AC=(0)^2-4(-1)(3)=12>0$

This equation represents a hyperbola.

Therefore the correct options are 2 and 5.

Answer 2

Try this option:
1. Common view of hyperbola equation is: 
$\frac{x^2}{a} - \frac{y^2}{b} =1 \ , where \ a,b-numbers$
2. hyperbolas are: 2x²+4x-5y²-10y+57=0 and -x²+12x+3y²+7y+11=0.